Finding Parameters of Axis Aligned Ellipse from Implicit Coordinate Equation

As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses, the Implicit Coordinate Equation for Axis Aligned Ellipses are given as

\(Ax^2 + Cy^2 + Dx + Ey +F=0\)   ...(1)

\(\Rightarrow Ax^2 + Dx + Cy^2 + Ey +F=0\)

\(\Rightarrow A(x^2 + \frac{D}{A}x) + C(y^2 + \frac{E}{C}y) +F=0\)   ...(2)

Completing the Squares in equation (2) we get

\(A(x^2 + 2\frac{D}{2A}x + \frac{D^2}{4A^2} - \frac{D^2}{4A^2}) + C(y^2 + 2\frac{E}{2C}y + \frac{E^2}{4C^2} - \frac{E^2}{4C^2}) + F =0\)

\(\Rightarrow A(x^2 + 2\frac{D}{2A}x + \frac{D^2}{4A^2}) + C(y^2 + 2\frac{E}{2C}y + \frac{E^2}{4C^2}) + F - \frac{D^2}{4A} - \frac{E^2}{4C} =0\)

\(\Rightarrow A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 + F - \frac{D^2}{4A} - \frac{E^2}{4C}=0\)

\(\Rightarrow A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 = \frac{D^2}{4A} + \frac{E^2}{4C} - F\)   ...(3)

Setting \(\frac{D^2}{4A} + \frac{E^2}{4C} - F = \mathbf{G}\) in equation (3) we get

\(A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 = G\)

\(\Rightarrow A{\Large \frac{{(x + \frac{D}{2A})}^2}{G}} + C{\Large \frac{{(y + \frac{E}{2C})}^2}{G}} = 1\)

\({\Rightarrow \Large\frac{{(x + \frac{D}{2A})}^2}{\frac{G}{A}}} + {\Large \frac{{(y + \frac{E}{2C})}^2}{\frac{G}{C}}} = 1\)   ...(4)

If equation (1) represents an Implicit Equation of Ellipse (Real or Imaginary), values of \({\Large \frac{G}{A}}\) and \({\Large \frac{G}{C}}\) given in equation (4) will have Same Sign.

If the values of \({\Large \frac{G}{A}} > 0\) and \({\Large \frac{G}{C}} > 0\), then the equation (1) represents a Real Ellipse and equation (4) gives it's Stardard Coordinate Equation.

If the values of \({\Large \frac{G}{A}} < 0\) and \({\Large \frac{G}{C}} < 0\), then the equation (1) represents an Imaginary Ellipse and equation (4) gives it's Stardard Coordinate Equation.
Using equation (4) Various Parameters of Ellipse given by Implicit Equation (1) can be given as
1. Coordinates of the Center: \((x_c,y_c)=(-\frac{D}{2A},-\frac{E}{2C})\)
2. Length of Semi-Major Axis: \(a=max(\sqrt{|\frac{G}{A}|},\sqrt{|\frac{G}{C}}|)\)
3. Length of Major Axis: \(L=2a\)
4. Length of Semi-Minor Axis: \(b=min(\sqrt{|\frac{G}{A}|},\sqrt{|\frac{G}{C}}|)\)
5. Length of Minor Axis: \(l=2b\)
6. Distance Between 2 Foci: \(F=\sqrt{L^2-l^2}=2c=2\sqrt{a^2-b^2}\)
7. Eccentricity \(e\) : \(\frac{F}{L}=\frac{c}{a}\)
8. Length for Latus Rectum : \(\frac{l^2}{L}=\frac{2b^2}{a}\)
9. If Major Axis is Parallel to \(X\)-Axis then
  Angular Orientation: 0°
  Coordinates of the 2 Foci: \((x_c + c,y_c),\hspace{.3cm}(x_c - c,y_c)\)
  Coordinates of the 2 Vertices: \((x_c + a,y_c),\hspace{.3cm}(x_c - a,y_c)\)
  Coordinates of the 2 Co-Vertices: \((x_c,y_c+b),\hspace{.3cm}(x_c,y_c-b)\)
  Equation of Major Axis: \(y=y_c\)
  Equation of Minor Axis: \(x=x_c\)
  Equation of 2 Directrices: \(x=x_c+\frac{L^2}{2F},\hspace{.3cm} x=x_c-\frac{L^2}{2F}\)   OR   \(x=x_c+\frac{a^2}{c},\hspace{.3cm} x=x_c-\frac{a^2}{c}\)
  Equation of 2 Latus Recta: \(x=x_c+c,\hspace{.5cm} x=x_c-c\)
  Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: \((x_c+c,y_c+\frac{l^2}{2L})\),\((x_c+c,y_c-\frac{l^2}{2L})\)   OR   \((x_c+c,y_c+\frac{b^2}{a})\),\((x_c+c,y_c-\frac{b^2}{a})\)
  Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: \((x_c-c,y_c+\frac{l^2}{2L})\),\((x_c-c,y_c-\frac{l^2}{2L})\)   OR   \((x_c-c,y_c+\frac{b^2}{a})\),\((x_c-c,y_c-\frac{b^2}{a})\)
10. If Major Axis is Parallel to \(Y\)-Axis then
  Angular Orientation: 90°
  Coordinates of the 2 Foci: \((x_c,y_c + c),\hspace{.3cm}(x_c,y_c - c)\)
  Coordinates of the 2 Vertices: \((x_c,y_c + a),\hspace{.3cm}(x_c,y_c - a)\)
  Coordinates of the 2 Co-Vertices: \((x_c+b,y_c),\hspace{.3cm}(x_c-b,y_c)\)
  Equation of Major Axis: \(x=x_c\)
  Equation of Minor Axis: \(y=y_c\)
  Equation of 2 Directrices: \(y=y_c+\frac{L^2}{2F},\hspace{.3cm} y=y_c-\frac{L^2}{2F}\)   OR   \(y=y_c+\frac{a^2}{c},\hspace{.3cm} y=y_c-\frac{a^2}{c}\)
  Equation of 2 Latus Recta: \(y=y_c+c,\hspace{.5cm} y=y_c-c\)
  Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: \((x_c+\frac{l^2}{2L},y_c+c)\),\((x_c-\frac{l^2}{2L},y_c+c)\)   OR   \((x_c+\frac{b^2}{a},y_c+c)\),\((x_c-\frac{b^2}{a},y_c+c)\)
  Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: \((x_c+\frac{l^2}{2L},y_c-c)\),\((x_c-\frac{l^2}{2L},y_c-c)\)   OR   \((x_c+\frac{b^2}{a},y_c-c)\),\((x_c-\frac{b^2}{a},y_c-c)\)

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